Conference Board of the Mathematical Sciences CBMS Regional Conference in

Number 110

Malliavin and Its Applications

David Nualart

American Mathematical Society with support from the National Science Foundation

Malliavin Calculus and Its Applications

http://dx.doi.org/10.1090/cbms/110

Conference Board of the Mathematical Sciences CBMS Regional Conference Series in Mathematics

Number 110

Malliavin Calculus and Its Applications

David Nualart

Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island with support from the National Science Foundation NSF-CBMS Regional Conference in the Mathematical Sciences on Malliavin Calculus and Its Applications held at Kent State University, Ohio, August 7–12, 2008 Partially supported by the National Science Foundation. The author acknowledges support from the Conference Board of Mathematical Sciences and NSF Grant #0735352. 2000 Mathematics Subject Classification. Primary 60H07; Secondary 60H05, 60H10.

For additional information and updates on this book, visit www.ams.org/bookpages/cbms-110

Library of Congress Cataloging-in-Publication Data Nualart, David, 1951– Malliavin calculus and its applications / David Nualart. p. cm. — (CBMS regional conference series in mathematics ; no. 110) Includes bibliographical references and index. ISBN 978-0-8218-4779-4 (alk. paper) 1. Malliavin calculus—Congresses. 2. —Congresses. I. Title. QA274.2 .N828 2009 519.2–dc22 2009003082

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Preface vii

Chapter 1. The operator 1 1.1. Hermite polynomials and chaos expansions 1 1.2. Derivative operator: Definition and properties 3 1.3. Derivative in the case 9

Chapter 2. The operator 13 2.1. Properties of the divergence 13 2.2. The divergence in the white noise case 15 2.3. Clark-Ocone formula 17

Chapter 3. The Ornstein-Uhlenbeck semigroup 19 3.1. Mehler’s formula 19 3.2. Hypercontractivity 20 3.3. The generator of the Ornstein-Uhlenbeck semigroup 23

Chapter 4. Sobolev spaces and equivalence of norms 27 4.1. Meyer inequalities 27 4.2. A continuous family of seminorms and Sobolev spaces 29 4.3. Continuity of the divergence 30

Chapter 5. Regularity of probability laws 33 5.1. Existence and formulas for probability densities 33 5.2. Regularity of the density 35

Chapter 6. Support properties. Density of the maximum 39 6.1. Properties of the support of the law 39 6.2. Regularity of the law of the maximum of continuous processes 40

Chapter 7. Application of Malliavin calculus to diffusion processes 45 7.1. Differentiability of the solution 46 7.2. Existence of densities under ellipticity conditions 48 7.3. Regularity of the density under H¨ormander’s conditions 49

Chapter 8. The divergence operator as a stochastic 55 8.1. Skorohod integral 55 8.2. for fractional 59

Chapter 9. Central limit theorems and Malliavin calculus 67 9.1. Central limit theorems via chaos expansions 67 9.2. Stein’s method and Malliavin calculus 72

v vi CONTENTS

Chapter 10. Applications of Malliavin calculus in finance 75 10.1. Black-Scholes model 75 10.2. Computation of Greeks 76 10.3. Application of the Clark-Ocone formula in hedging 80 Bibliography 81 Index 85 Preface

This monograph is compiled from the notes of a series of ten lectures given at the NSF-CBMS Conference on Malliavin Calculus and Its Applications at Kent State University, Ohio, August 7th to 12th, 2008. The Malliavin calculus or stochastic calculus of variations is an infinite-dimen- sional differential calculus on the Wiener space that has been developed from the probabilistic proof of H¨ormander’s hypoellipticity theorem by Paul Malliavin in 1976 (see the reference [25]). Contributions by Stroock, Bismut, Kusuoka, and Watanabe, among others, have expanded this theory in different directions. The main application of Malliavin calculus is to establish the regularity of the of functionals of an underlying . In this way one can prove the existence and smoothness of the density for solutions to ordinary and partial stochastic differential equations. In addition to this main application, the Malliavin calculus has proved to be a powerful tool in a variety of problems in stochastic . For example, the divergence operator can be interpreted as a generalized stochastic integral, and this has been the starting point of the development of the anticipating stochastic calculus. In the last years some new applications of Malliavin calculus in areas such as central limit theorems and mathematical finance have emerged. The purpose of these lectures is to introduce the basic results of Malliavin calculus and its applications. We have chosen the general setting of a Gaussian family of random variables associated with an arbitrary separable Hilbert space. Some of the applications are just briefly introduced, and we recommend the reader to look over additional references for more details. In the first three chapters we introduce the fundamental operators: the deriva- tive operator D;itsadjointδ, called the divergence operator; and the generator of the Ornstein-Uhlenbeck semigroup, denoted by L. Chapter 4 is devoted to proving the Meyer inequalities and the continuity of the divergence operator in the Sobolev spaces. The remaining chapters deal with a variety of applications of Malliavin calculus. First, in Chapter 5, we establish the general criteria for the existence and smoothness of densities for functionals of a Gaussian process. In Chapter 6 we discuss properties of the support of the law of a given Gaussian functional that can be proved using Malliavin calculus. Chapter 7 deals with the proof of H¨ormander’s hypoellipticity theorem. In Chapter 8 we discuss the use of the divergence as an anticipating stochastic integral with respect to the Brownian motion. This chapter also contains an introduction to the stochastic calculus with respect to the fractional Brownian motion, using techniques of Malliavin calculus. Chapter 9 presents some recent applications of Malliavin calculus to derive central limit theorems for mul- tiple stochastic , and Chapter 10 describes some applications of Malliavin calculus in mathematical finance.

vii viii PREFACE

Finally, I would like to express my gratitude to Oana Mocioalca, Frederi Viens, and Kazim Khan for encouraging me to prepare these lectures and for organizing a very stimulating and interesting workshop. I would also like to thank all the participants for their helpful remarks and questions.

David Nualart Bibliography

[1] E. Al`os, O. Mazet and D. Nualart, Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001), 766–801. [2] E. Al`os and D. Nualart, An extension of Itˆo’s formula for anticipating processes. J. Theoret. Probab. 11 (1998), 493–514. [3] D. Bakry, L’hypercontractivit´e et son utilisation en th´eorie des semigroupes. Lectures on (Saint-Flour, 1992), Lecture Notes in Math., 1581, Springer, Berlin, 1994, pp. 1-114. [4] D. Bell, The Malliavin Calculus, Dover Publications Inc., Mineola, New York, 2006. [5] F. Biagini, Y. Hu, B. Øksendal, T. Zhang, Stochastic calculus for fractional Brownian motion and applications. Probability and its Applications, Springer, London, 2008. [6] J. M. Bismut, Martingales, the Malliavin calculus and hypoellipticity under general Hrman- der’s conditions. Z. Wahrsch. Verw. Gebiete 56 (1981), 469–505. [7] J. M. Bismut, Large deviations and the Malliavin calculus, Progress in Math. 45, Birkh¨auser, 1984. [8] N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space,WalterdeGruyter & Co., Berlin, 1991. [9] Ph. Carmona, L. Coutin, and G. Montseny, Stochastic integration with respect to fractional Brownian motion. Ann. Inst. H. Poincar´e 39 (2003), 27–68. [10] P. Cheridito and D. Nualart, Stochastic integral of divergence type with respect to the frac- H<1 41 tional Brownian motion with Hurst parameter 2 . Ann. Inst. H. Poincar´e (2005), 1049–1081. [11] J. M. Corcuera, D. Nualart and J. H. C. Woerner, Power variation of some integral fractional processes. Bernoulli 12 (2006), 713–735. [12] L. Decreusefond and A. S. Ust¨¨ unel, Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1998), 177–214. [13] E. Fourni´e, Eric, J. M. Lasry, J. Lebuchoux, P. L. Lions and Nizar Touzi, Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch. 3 (1999), 391–412. [14] B. Gaveau and P. Trauber, L’int´egrale stochastique comme op´erateur de divergence dans l’espace fonctionnel. J. Functional Anal. 46 (1982), 230–238. [15] Gross, Abstract Wiener spaces. 1967 Proc. Fifth Berkeley Sympos. Math. Statist. and Prob- ability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1 pp. 31–42, Univ. California Press, Berkeley, Calif. [16] Y. Hu and D. Nualart, Renormalized self-intersection for fractional Brownian motion. Ann. Probab. 33 (2005), 948–983. [17] K. Itˆo, Multiple Wiener integral. J. Math. Soc. Japan 3 (1951), 157–169. [18] D. L. Ocone and I. Karatzas, Generalized Clark representation formula, with application to optimal portfolios. Stochastics Stochastics Rep. 34 (1991), 187–220. [19] A. Kohatsu-Higa and M. Montero, Malliavin calculus in finance. Handbook of computational and numerical methods in finance, Birkh¨auser, Boston, 2004, pp. 111–174. [20] A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C. R. (Doklady) Acad. URSS, 26 (1940), 115–118. [21] H. Kunita, Stochastic differential equations and stochastic flow of diffeomorphisms. Ecole d’Et´e de Probabilit´es de Saint Flour XII, 1982, Lecture Notes in Math. 1097, Springer, Berlin, 1984, pp. 144–305. [22] S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. I. Stochastic analysis (Katata/Kyoto, 1982), North-Holland, Amsterdam, 1984, pp. 271–306.

81 82 BIBLIOGRAPHY

[23] S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), 1–76. [24] S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), 391–442. [25] P. Malliavin, Stochastic calculus of variation and hypoelliptic operators. Proceedings of the International Symposium on Stochastic Differential Equations, Kyoto, 1976, Wiley, New York-Chichester-Brisbane, 1978, pp. 195–263. [26] P. Malliavin, Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften, 313, Springer, Berlin, 1997. [27] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications. SIAM Review 10 (1968), 422–437. [28] P. A. Meyer, Transformations de Riesz pour les lois gaussiennes. Seminaire de Probabilit´es XVIII, Lecture Notes in Math. 1059, 1984, pp. 179–193. [29] J. Memin, Y. Mishura and E. Valkeila, Inequalities for the moments of Wiener integrals with respecto to fractional Brownian motions. Statist. Prob. Letters 55 (2001), 421–430. [30] J. Neveu, Sur l’esp´erance conditionnelle par rapporta ` un mouvement brownien. Ann. Inst. H. Poincar´e 12 (1976), 105–109. [31] J. Norris., Simplified Malliavin calculus. S´eminaire de Probabilit´es, XX, 1984/85, Lecture Notes in Math. 1204, 1986, pp. 101–130. [32] I. Nourdin, G. Peccati, Stein’s method on Wiener chaos. Preprint. [33] I. Nourdin, G. Peccati, Stein’s method and exact Berry-Ess´een asymptotics for functionals of Gaussian fields. Preprint. [34] D. Nualart, Analysis on Wiener space and anticipating stochastic calculus. Lectures on prob- ability theory and (Saint-Flour, 1995), Lecture Notes in Math. 1690, 1998, pp. 123–227. [35] D. Nualart, The Malliavin Calculus and Related Topics. Second edition, Springer , New York, 2006. [36] D. Nualart, Application of Malliavin calculus to stochastic partial differential equations. A Minicourse on Stochastic Partial Differential Equations, Lecture Notes in Math. 1962, Springer, Berlin, 2008, pp. 73–109. [37] D. Nualart and S. Ortiz-Latorre, Central limit theorems for multiple stochastic integrals. Stochastic Processes Appl. 118 (2007), 614–628. [38] D. Nualart and E. Pardoux, Stochastic calculus with anticipating integrands. Probab. Theory Rel. Fields 78 (1988), 535–581. [39] D. Nualart and G. Peccati, entral limit theorems for sequences of multiple stochastic integrals. Ann. of Probab. 33 (2005), 177-193 [40] D. Nualart and M. Zakai, Generalized stochastic integrals and the Malliavin calculus. Probab. Theory Rel. Fields 73 (1986), 255–280. [41] D. Nualart and M. Zakai., Generalized multiple stochastic integrals and the representation of Wiener functionals. Stochastics23 (1988), 311-330. [42] E. Pardoux and P. Protter, Two-sided stochastic integrals and calculus. Probab. Theory Rel. Fields 76 (1987), 15–50. [43] G. Peccati and C. Tudor, Gaussian limits for vector-valued multiple stochastic integrals. S´eminaire de Probabilit´es XXXVIII, Lecture Notes in Math., 1857, Springer, Berlin, 2005, pp. 247-262. [44] V. Pipiras and M. S. Taqqu, Are classes of deterministic integrands for fractional Brownian motion on a interval complete? Bernoulli 7 (2001), 873–897 [45] G. Pisier, Riesz transforms. A simple analytic proof of P. A. Meyer’s inequality. Seminaire de Probabilit´es XXIII, Lecture Notes in Math. 1321, (1988, pp. 485–501. [46] A. V. Skorohod, On a generalization of a stochastic integral. Theory Probab. Appl. 20 (1975), 219–233. [47] D. W. Stroock, Some applications of stochastic calculus to partial differential equations. Eleventh Saint Flour probability summer school 1981 (Saint Flour, 1981), Lecture Notes in Math., 976, Springer, Berlin, 1983, pp. 267–382. [48] D. W. Stroock, Homogeneous chaos revisited. Seminaire de Probabilit´es XXI, Lecture Notes in Math. 1247, 1987, pp. 1–8. [49] E. M. Stein, Singular Integrals and Differentiability of Functions. Princeton Univ. Press, 1970. BIBLIOGRAPHY 83

[50] A. S. Ust¨¨ unel, An introduction to analysis on Wiener space.Lecture Notes in Mathematics, 1610, Springer-Verlag, Berlin, 1995. [51] S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus. Tata Institute of Fundamental Research, Springer-Verlag, 1984. [52] L. C. Young, An inequality of the H¨older type connected with Stieltjes integration. Acta Math. 67 (1936), 251–282.

Index

abstract Wiener space, 45 Mehler’s formula, 19 anticipating stochastic integrals, 55 Meyer inequalities, 27 multiple stochastic integrals, 9 Black-Scholes model, 75 Brownian motion, 1 nondegenerate random vector, 35 Brownian sheet, 42 Norris lemma, 50

Cameron-Martin space, 45 Ornstein-Uhlenbeck process, 20 Cauchy semigroup, 25 Ornstein-Uhlenbeck semigroup, 19 central limit theorem, 67 , 5 Picard approximations, 46 chaotic central limit theorem, 70 progressively measurable process, 16 Clark-Ocone formula, 17 reduced Malliavin matrix, 48 contraction of order r,67 reproducing kernel Hilbert space, 42 covariant derivative, 49 Riemansn sums, 57

Delta, 77 slef-financing portfolio, 75 derivative operator, 3 self-similarity, 59 difussion process, 45 Skorohod integral, 55 divergence operator, 13 smooth and cylindrical random variables, 3 duality relationship, 13 Sobolev spaces, 29 ellipticity conditions, 48 Stein’s equation, 72 European options, 77 Stein’s method, 72 stochastic differential equation, 45 forward stochastic integral, 58 , 49 fractional Brownian motion, 59 Stroock’s formula, 10 support of the law, 39 Gamma, 77 Gaussian white noise, 1 two-parameter , 42 Greeks, 76 Vega, 77 integration-by-parts formula, 3 volatility, 75 Itˆo stochastic integral, 16 nth Wiener chaos, 3 Hermite Polynomials, 1 Wiener chaos expansion, 6 H¨ormander’s conditions, 49 Wiener measure, 45 H¨ormander’s theorem, 50 hypercontractivity, 20

Lie bracket, 49 local property of the derivative, 40 local property of the divergence, 56 local time, 18

Malliavin matrix, 33 mean return rate, 75

85

Titles in This Series

110 David Nualart, Malliavin calculus and its applications, 2009 109 Robert J. Zimmer and Dave Witte Morris, Ergodic theory, groups, and geometry, 2008 108 Alexander Koldobsky and Vladyslav Yaskin, The interface between convex geometry and , 2008 107 FanChungandLinyuanLu, Complex graphs and networks, 2006 106 Terence Tao, Nonlinear dispersive equations: Local and global analysis, 2006 105 Christoph Thiele, Wave packet analysis, 2006 104 Donald G. Saari, Collisions, rings, and other Newtonian N-body problems, 2005 103 Iain Raeburn, Graph algebras, 2005 102 Ken Ono, The web of modularity: Arithmetic of the coefficients of modular forms and q series, 2004 101 Henri Darmon, Rational points on modular elliptic curves, 2004 100 Alexander Volberg, Calder´on-Zygmund capacities and operators on nonhomogeneous spaces, 2003 99 Alain Lascoux, Symmetric functions and combinatorial operators on polynomials, 2003 98 Alexander Varchenko, Special functions, KZ type equations, and representation theory, 2003 97 Bernd Sturmfels, Solving systems of polynomial equations, 2002 96 Niky Kamran, Selected topics in the geometrical study of differential equations, 2002 95 Benjamin Weiss, Single orbit dynamics, 2000 94 David J. Saltman, Lectures on division algebras, 1999 93 Goro Shimura, Euler products and Eisenstein series, 1997 92 FanR.K.Chung, Spectral , 1997 91 J. P. May et al., Equivariant homotopy and cohomology theory, dedicated to the memory of Robert J. Piacenza, 1996 90 John Roe, Index theory, coarse geometry, and topology of manifolds, 1996 89 Clifford Henry Taubes, Metrics, connections and gluing theorems, 1996 88 Craig Huneke, Tight closure and its applications, 1996 87 John Erik Fornæss, Dynamics in several complex variables, 1996 86 Sorin Popa, Classification of subfactors and their endomorphisms, 1995 85 Michio Jimbo and Tetsuji Miwa, Algebraic analysis of solvable lattice models, 1994 84 Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, 1994 83 Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, 1994 82 Susan Montgomery, Hopf algebras and their actions on rings, 1993 81 Steven G. Krantz, and function spaces, 1993 80 Vaughan F. R. Jones, Subfactors and knots, 1991 79 Michael Frazier, Bj¨orn Jawerth, and Guido Weiss, Littlewood-Paley theory and the study of function spaces, 1991 78 Edward Formanek, The polynomial identities and variants of n × n matrices, 1991 77 Michael Christ, Lectures on singular integral operators, 1990 76 Klaus Schmidt, Algebraic ideas in ergodic theory, 1990 75 F. Thomas Farrell and L. Edwin Jones, Classical aspherical manifolds, 1990 74 Lawrence C. Evans, Weak convergence methods for nonlinear partial differential equations, 1990 73 Walter A. Strauss, Nonlinear wave equations, 1989 TITLES IN THIS SERIES

72 Peter Orlik, Introduction to arrangements, 1989 71 Harry Dym, J contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, 1989 70 Richard F. Gundy, Some topics in probability and analysis, 1989 69 Frank D. Grosshans, Gian-Carlo Rota, and Joel A. Stein, Invariant theory and , 1987 68 J. William Helton, Joseph A. Ball, Charles R. Johnson, and John N. Palmer, , analytic functions, matrices, and electrical engineering, 1987 67 Harald Upmeier, Jordan algebras in analysis, operator theory, and quantum mechanics, 1987 66 G. Andrews, q-Series: Their development and application in analysis, number theory, , physics and , 1986 65 Paul H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, 1986 64 Donald S. Passman, Group rings, crossed products and Galois theory, 1986 63 Walter Rudin, New constructions of functions holomorphic in the unit ball of Cn, 1986 62 B´ela Bollob´as, Extremal graph theory with emphasis on probabilistic methods, 1986 61 Mogens Flensted-Jensen, Analysis on non-Riemannian symmetric spaces, 1986 60 Gilles Pisier, Factorization of linear operators and geometry of Banach spaces, 1986 59 Roger Howe and Allen Moy, Harish-Chandra homomorphisms for p-adic groups, 1985 58 H. Blaine Lawson, Jr., The theory of gauge fields in four dimensions, 1985 57 Jerry L. Kazdan, Prescribing the curvature of a Riemannian manifold, 1985 56 Hari Bercovici, Ciprian Foia¸s, and Carl Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, 1985 55 William Arveson, Ten lectures on operator algebras, 1984 54 William Fulton, Introduction to intersection theory in algebraic geometry, 1984 53 Wilhelm Klingenberg, Closed geodesics on Riemannian manifolds, 1983 52 Tsit-Yuen Lam, Orderings, valuations and quadratic forms, 1983 51 Masamichi Takesaki, Structure of factors and automorphism groups, 1983 50 James Eells and Luc Lemaire, Selected topics in harmonic maps, 1983 49 John M. Franks, Homology and dynamical systems, 1982 48 W. Stephen Wilson, Brown-Peterson homology: an introduction and sampler, 1982 47 Jack K. Hale, Topics in dynamic bifurcation theory, 1981 46 Edward G. Effros, Dimensions and C∗-algebras, 1981 45 Ronald L. Graham, Rudiments of Ramsey theory, 1981 44 Phillip A. Griffiths, An introduction to the theory of special divisors on algebraic curves, 1980 43 William Jaco, Lectures on three-manifold topology, 1980 42 Jean Dieudonn´e, Special functions and linear representations of Lie groups, 1980 41 D. J. Newman, Approximation with rational functions, 1979 40 Jean Mawhin, Topological degree methods in nonlinear boundary value problems, 1979 39 George Lusztig, Representations of finite Chevalley groups, 1978 38 Charles Conley, Isolated invariant sets and the Morse index, 1978 37 Masayoshi Nagata, Polynomial rings and affine spaces, 1978

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. The Malliavin calculus was developed to provide a probabilistic proof of Hörmander’s hypoellipticity theorem. The theory has expanded to encompass other significant applications. The main application of the Malliavin calculus is to establish the regularity of the probability distribution of functionals of an underlying Gaussian process. In this way, one can prove the existence and smoothness of the density for solutions of various stochastic differential equations. More recently, applications of the Malliavin calculus in areas such as stochastic calculus for fractional Brownian motion, central limit theo- rems for multiple stochastic integrals, and have emerged. The first part of the book covers the basic results of the Malliavin calculus. The middle part establishes the existence and smoothness results that then lead to the proof of Hörmander’s hypoellipticity theorem. The last part discusses the recent developments for Brownian motion, central limit theorems, and mathematical finance.

For additional information and updates on this book, visit www.ams.org/bookpages/cbms-110

AMS on the Web CBMS/110 www.ams.org